Gross atomic orbital population

As tne data in Table 6 show, the sum of the terms neglected in the Townes and Dailey method is by no means small. Cotton and Harris 3,) have attempted to re-vamp the method and after a series of approximations which at first sight appear more reasonable than the original ones, arrive at a series of equations in which the valence orbital populations of the Townes and Dailey method are replaced by the corresponding Mulliken gross atomic orbital populations. 

One final definition of note is the gross atomic orbital population NfrJ, Eq. [13], which is the number of electrons in a particular atomic orbital. 

The (PS)., elements are the gross atomic orbital population of x.- The gross atomic population is then... 

Mulliken also noted another disturbing tendency in this population anal-ysis. - The gross atomic orbital population could have a value less than zero or greater than 2. A population greater than 2 appears to violate at least the spirit of the Pauli exclusion principle. A negative orbital population is even more difficult to interpret. Mulliken s examples all had very small negative values and were thus discounted as unimportant. However, the atomic population can be very negative—a population of —0.7 e for a sulfur d orbital in SFg... 

Note that this equation has no cross terms. The gross atomic orbital population is simply a sum over all MOs of the orbital coefficient squared, which means that only positive values will be obtained. Since the wavefunction is normalized, the atomic orbital population can be no larger than N(i), which is usually 2. Gross atomic populations are obtained by summing the atomic orbital population of all orbitals centered on a given atom. 

Here, Q- is the partial overlap population, namely, the electron population of the overlap region between the atomic orbitals y,- and Xj >n the MO atomic orbitals y,- and y, in the MO , and S,(, the overlap integral between atomic orbitals y,- and y . QH is the gross atomic population or the gross atomic charge on atom H and is given by the sum of the atomic orbital populations Qi. The net charge AQh is obtained from the difference between QH and the atomic number ZH, namely, the number of electrons in the neutral atom. 

The population method due to Pipek and Mezey [37] maximizes a certain sum of gross atomic Mulliken populations. The latter are not realistic when the LCAO basis includes diffuse atomic orbitals as is necessary for crystalhne sohds. 

According to molecular orbital calculations, the interaciion is mainly electrostatic in character. Only minor variations of the gross atomic population in the associate, as compared with the isolated molecule, are found Consistent with this view, the experimental stretching frequency of the CN bond of organic nitriles in the vapor phase (where supposedly the monomeric form predominates) and in liquid state (where 70—90% are associated) are very amilar, as shown in Table 2. If the small decrease of 15—20 cm is taken as an indication for dimer formation, it may be conduded that the degree of association is similar in the pure liquid and in CCI4 solution. 

In order to obtain information concerning the bonding nature, we used the Mulliken population analysis (8). The number of electrons are partitioned into gross populations for the rth atomic orbital, n,-... 

A recent comparison (33.) of quantities calculated from molecular electronic wave functions for pyrrole and pyrazole indicated that ARCANA values compared more favorably with large-scale ab initio values than those calculated by other non-rigorous methods in the case of orbital energies of occupied molecular orbitals, gross atomic populations of heteroatoms, and total overlap populations including negative overlap populations between nonbonded atoms. 

If both the M0DP0T and VRDD0 approximations are introduced, the accuracy [ see footnote page Ul9] with respect to the reference calculations is about the same as that obtained using only the M0DP0T approximation. The maximum error in the orbital energies is 0.012 a.u., in the gross atomic populations is 0.009> in the total overlap populations is 0.027 and in the dipole moment is 0.009 a.u. 

Alternatively, Doggett suggested weighting the contribution of the overlap population by the position of the orbital centroid, relative to the midpoint of the bond. In this scheme, the gross atomic population is... 

Davidson suggested that the wavefunction be projected onto a set of orbitals that have intuitive significance. These orbitals are a minimum set of atomic orbitals that provide the best least-squares fit of the first-order reduced-density matrix. Roby expanded on this idea by projecting onto the wavefunction of the isolated atom. One then uses the general Mulliken idea of counting the number of electrons in each of these projected orbitals that reside on a given atom to obtain the gross atomic population. 

Table 3.7. Surfac atomic orbital gross populations twofold adsorption of CO on Rh(lll).

Tnble 3.17. Gross population of Hartree-SNter-Fock Linear Combination of Atomic Orbital calculationi ) ). 

Here the indices A and B refer to atoms, k and I label atomic orbitals centered on atoms A and B, respectively, and i labels the molecular orbitals in the SCF approximation - which may be either doubly or singly occupied depending on the RHF or UHF case, respectively c,vt represents the coefficient of atomic orbital k in the molecular orbital i and Su is the overlap integral between atomic orbitals k and 1. The charge of the atom appears by comparison between the gross population Pa and the atomic number Z ... 

To elucidate the role of f-type polarization functions on describing chemical bonds, natural bond orbital (NBO) analyses were carried out for the computational results of method ii. For the NBOs of S=0 bonds in all six derivatives, the contributions of f-type orbitals were largest of all 2-center bond NBOs. The second largest contributions of f-type orbitals were observed in NBOs of S-N bond. Both for S=0 and S-N, the contributions of f-type orbital of sulfur atoms were much larger than those of oxygen or nitrogen atoms. In addition, gross orbital populations of f-type... 

The gross atomic population, which gives a measure of the formal charge of an atom in a molecule, is defined as the sum over all N(r) for r a spin orbital on the atom under consideration. 

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